**講義名 英語理学講義（数学２）（ Science in English (Mathematics2) ）
開講学期 後学期 単位数 1--0--0
担当 Andrei Pajitnov 東京工業大学理学研究流動機構客員教授 （フランス・ナント大学教授）
【講義題目】**

Circle-valued Morse theory and Novikov homology

【講義内容】

1) Novikov complex for circle-valued Morse functions.

2) Witten-type de Rham framework for Novikov inequalities.

3) Morse-Novikov theory for 3-knots

4) Applications to symplectic topology.

Classical Morse theory establishes a relation between the number

of critical points of a real-valued function on a manifold and the

topology of the manifold.

Circle-valued Morse theory is a branch of the Morse theory.

It originated from a problem in hydrodynamics studied by

S. Novikov in the early 1980s.

Nowadays it is a constantly growing field of geometry with

applications and connections to many geometric problems

such as Arnold's conjectures in symplectic topology,

fibrations of manifolds over the circle, dynamical zeta functions,

and the theory of 3-dimensional knots and links.

Our course will start with recollections on the classical Morse

theory (gradient flows, Morse complex, Morse inequalities).

Then we proceed to the circle-valued Morse theory and

construct the Novikov complex (the generalization of the Morse

complex to the case of circle-valued functions).

This chain complex is generated over the Laurent series ring

by the critical points of the circle-valued function.

E. Witten showed in the beginning of 1980s that the classical Morse

theory can be reformulated in the framework of the de Rham theory.

We will explain his work, and its generalization to the Morse-Novikov

theory, which leads to a relation between the Novikov homology

and the homology with local coefficients.

A natural application of the circle-valued Morse theory is to the

topology of knots. If a knot K in the 3-dimensional sphere S is not

fibred, then any circle-valued function on S-K has critical points.

The minimal number of these critical points is an invariant of K,

called the Morse-Novikov number MN(K).

We will show how to give computable lower bounds for MN(K)

with the help of the Novikov homology, and relate it to the tunnel

number of the knot.

In the last part of the course we will explain how the Novikov ring

appears in the Floer's work on the Arnold conjecture concerning

the closed orbits of periodic Hamiltonians, and discuss some recent

developments in this direction.

２０１５年１月２６日(月)〜１月３０日(金)

７、８時限（１５：０５～１６：３５）

本館２階２０１セミナー室

評価方法については講義中に指示する。